A multiple integral is a set of integrals taken over variables, e.g.,
An th-order
integral corresponds, in general, to an -dimensional volume (i.e., a content), with corresponding to an area. In an
indefinite multiple integral, the order in which the integrals are carried out can
be varied at will; for definite multiple integrals, care must be taken to correctly
transform the limits if the order is changed.
In traditional mathematical notation, a multiple integral of a function that is first performed over a variable and then performed over a variable is written
In the Wolfram Language, this would be entered as Integrate[f[x,
y], x,
x1, x2,
y,
y1[x], y2[x]], where the order of the integration variables is specified
in the order that the integral signs appear on the left, which is opposite to
the actual order of integration.
Berntsen, J.; Espelid, T. O.; and Genz, A. "An Adaptive Algorithm for the Approximate Calculation of Multiple Integrals." ACM
Trans. Math. Soft.17, 437-451, 1991.Kaplan, W. "Double
Integrals" and "Triple Integrals and Multiple Integrals in General."
§4.3-4.4 in Advanced
Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 228-235, 1991.Press,
W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T.
"Multidimensional Integrals." §4.6 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 155-158, 1992.
variables, e.g.,
th-order
integral corresponds, in general, to an 
-dimensional volume (i.e., a content), with 
corresponding to an area. In an
indefinite multiple integral, the order in which the integrals are carried out can
be varied at will; for definite multiple integrals, care must be taken to correctly
transform the limits if the order is changed.
that is first performed over a variable 
and then performed over a variable 
is written
![int_(x_1)^(x_2)[int_(y_1(x))^(y_2(x))f(x,y)dy]dx=int_(x_1)^(x_2)int_(y_1(x))^(y_2(x))f(x,y)dydx.](/images/equations/MultipleIntegral/NumberedEquation2.svg)
x,
x1, x2
,
y,
y1[x], y2[x]
], where the order of the integration variables is specified
in the order that the integral signs appear on the left, which is opposite to
the actual order of integration.
